We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which (...) may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987. (shrink)

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory T is not $\omega $ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct.

We show, for any ordinal γ ≥ 3, that the class RaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fⁿ (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra A with countably many atoms, that Ǝ has a winning strategy in Fω(At(A)) ⇔ A ∈ ScRaCAω, Ǝ has a winning strategy in Fⁿ(At(A)) ⇐ A ∈ ScRaCAn, Ǝ (...) has a winning strategy in G(At(A)) ⇐ A ∈ RaCAω, Ǝ has a winning strategy in H(At(A)) ⇒ A ∈ RaRCAω for 3 ≤ n < ω. We use these games to show, for γ ≥ 5 and any class K of relation algebras satisfying RaRCAγ ⊆ K ⊆ ScRaCA₅, that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion RaCAγ ⊂ ScRaCAγ is strict. For infinite γ and for a countable relation algebra A we show that A has a complete representation if and only if A is atomic and Ǝ has a winning strategy in F(At(A)) if and only if A is atomic and A ∈ ScRaCAγ. (shrink)

Following research initiated by Tarski, Craig and Nemeti, and futher pursued by Sain and others, we show that for certain subsets G of $^\omega \omega $ , atomic countable G poiyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras.

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.

Given two structures${\cal M}$and${\cal N}$on the same domain, we say that${\cal N}$is a reduct of${\cal M}$if all$\emptyset$-definable relations of${\cal N}$are$\emptyset$-definable in${\cal M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are${\aleph _0}$-categorical, determining their reducts is equivalent to determining the closed supergroupsG≤ Sym of their automorphism groups.A consequence of the classification is that there are${2^{{\aleph _0}}}$pairwise noninterdefinable Henson digraphs (...) which have no proper nontrivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are${2^{{\aleph _0}}}$pairwise nonconjugate maximal-closed subgroups of Sym. By the reconstruction results of Rubin, these groups are also nonisomorphic as abstract groups. (shrink)

Reduction and reductionism have been central philosophical topics in analytic philosophy of science for more than six decades. Together they encompass a diversity of issues from metaphysics and epistemology. This article provides an introduction to the topic that illuminates how contemporary epistemological discussions took their shape historically and limns the contours of concrete cases of reduction in specific natural sciences. The unity of science and the impulse to accomplish compositional reduction in accord with a layer-cake vision of the sciences, the (...) seminal contributions of Ernest Nagel on theory reduction and how they strongly conditioned subsequent philosophical discussions, and the detailed issues pertaining to different accounts of reduction that arise in both physical and biological science (e.g., limit-case and part-whole reduction in physics, the difference-making principle in genetics, and mechanisms in molecular biology) are explored. The conclusion argues that the epistemological heterogeneity and patchwork organization of the natural sciences encourages a pluralist stance about reduction. (shrink)

A conventional wisdom about the progress of physics holds that successive theories wholly encompass the domains of their predecessors through a process that is often called reduction. While certain influential accounts of inter-theory reduction in physics take reduction to require a single "global" derivation of one theory's laws from those of another, I show that global reductions are not available in all cases where the conventional wisdom requires reduction to hold. However, I argue that a weaker "local" form of reduction, (...) which defines reduction between theories in terms of a more fundamental notion of reduction between models of a single fixed system, is available in such cases and moreover suffices to uphold the conventional wisdom. To illustrate the sort of fixed-system, inter-model reduction that grounds inter-theoretic reduction on this picture, I specialize to a particular class of cases in which both models are dynamical systems. I show that reduction in these cases is underwritten by a mathematical relationship that follows the broad prescriptions of Nagel/Schaffner reduction, and support this claim with several examples. Moreover, I show that this broadly Nagelian analysis of inter-model reduction encompasses several cases that are sometimes cited as instances of the "physicist's" limit-based notion of reduction. (shrink)

Ethicists struggle to take reductive views seriously. They also have trouble conceiving of some supervenience failures. Understanding why provides further evidence for a kind of hybrid view of normative concept use.

Approaches to the naturalization of phenomenology usually understand naturalization as a matter of rendering continuous the methods, epistemologies, and ontologies of phenomenological and natural scientific inquiry. Presupposed in this statement of the problematic, however, is that there is an original discontinuity, a rupture between phenomenology and the natural sciences that must be remedied. I propose that this way of thinking about the issue is rooted in a simplistic understanding of the phenomenological reduction that entails certain assumptions about the subject matter (...) of phenomenology and its relationship to the natural sciences. By contrast, Merleau‐Ponty's first work, The Structure of Behavior, presents a radically different approach to the phenomenological reduction, one that traverses the natural sciences and integrates them into phenomenology from the outset. I outline the argument for this position in The Structure of Behavior and then discuss consequences for current methodological issues surrounding the naturalization of phenomenology, focusing on the relationship between empirical sciences of mind, phenomenological psychology, and transcendental phenomenology. This novel exegesis of Merleau‐Ponty's view on the reduction offers new insight into his oft‐quoted remark that the phenomenological reduction is impossible to complete. (shrink)

Rudolf Carnap proposed three reduction rules to improve scientific language. Those rules indicate when it is correct to add a property in situations in which other properties are already had. Assuming the theory of mental models, it has been shown that the rules are not hard to use from the cognitive point of view. The key to argue that is to accept that the human mind works as dual-process theories claim. According to these theories, people can use two different systems. (...) One of them implies effort but the other one does not. Thus, the idea is that Carnap’s reduction rules can be applied resorting just to the system not implying cognitive effort. This paper goes one step further and poses that even from a Popperian perspective the reduction rules keep not being difficult. One might think that falsifiability requires cognitive effort because it needs to address situations in which the sentences are false. However, this paper tries to explain that, following the theory of mental models, that is not hard if the sentences are conditionals, and the sentences in the reduction rules have conditional structures. (shrink)

In a recent critique of the doctrine of emergentism championed by its classic advocates up to C. D. Broad, Jaegwon Kim (Philosophical Studies 63:31–47, 1999) challenges their view about its applicability to the sciences and proposes a new account of how the opposing notion of reduction should be understood. Kim is critical of the classic conception advanced by Nagel and uses his new account in his criticism of emergentism. I question his claims about the successful reduction achieved in the sciences (...) and argue that his new account has not improved on Nagel’s and that the critique of emergentism he bases on it is question-begging in important respects. (shrink)

I argue that an adequate account of non-reductive realization must guarantee satisfaction of a certain condition on the token causal powers associated with (instances of) realized and realizing entities---namely, what I call the 'Subset Condition on Causal Powers' (first introduced in Wilson 1999). In terms of states, the condition requires that the token powers had by a realized state on a given occasion be a proper subset of the token powers had by the state that realizes it on that occasion. (...) Accounts of non-reductive realization conforming to this condition are implementing what I call 'the powers-based subset strategy'. I focus on the crucial case involving mental and brain states; the results may be generalized, as appropriate. I first situate and motivate the strategy by attention to the problem of mental causation; I make the case, in schematic terms, that implementation of the strategy makes room (contra Kim 1989, 1993, 1998, and elsewhere) for mental states to be ontologically and causally autonomous from their realizing physical states, without inducing problematic causal overdetermination, and compatible with both Physicalism and Non-reduction; and I show that several contemporary accounts of non-reductive realization (in terms of functional realization, parthood, and the determinable/determinate relation) are plausibly seen as implementing the strategy. As I also show, implementation of the powers-based strategy does not require endorsement of any particular accounts of either properties or causation---indeed, a categoricalist contingentist Humean can implement the strategy. The schematic location of the strategy in the space of available responses to the problem of mental (more generally, higher-level) causation, as well as the fact that the schema may be metaphysically instantiated, strongly suggests that the strategy is, appropriately generalized and instantiated, sufficient and moreover necessary for non-reductive realization. I go on to defend the sufficiency and necessity claims against a variety of objections, considering, along the way, how the powers-based subset strategy fares against competing accounts of purportedly non-reductive realization in terms of supervenience, token identity, and constitution. (shrink)

Some claim that Non-reductive Physicalism is an unstable position, on grounds that NRP either collapses into reductive physicalism, or expands into emergentism of a robust or ‘strong’ variety. I argue that this claim is unfounded, by attention to the notion of a degree of freedom—roughly, an independent parameter needed to characterize an entity as being in a state functionally relevant to its law-governed properties and behavior. I start by distinguishing three relations that may hold between the degrees of freedom needed (...) to characterize certain special science entities, and those needed to characterize their composing physical entities; these correspond to what I call ‘reductions’, ‘restrictions’, and ‘eliminations’ in degrees of freedom. I then argue that eliminations in degrees of freedom, in particular—when strictly fewer degrees of freedom are required to characterize certain special science entities than are required to characterize their composing physical entities—provide a basis for making sense of how certain special science entities can be both physically acceptable and ontologically irreducible to physical entities. (shrink)

Philosophers of neuroscience have traditionally described interfield integration using reduction models. Such models describe formal inferential relations between theories at different levels. I argue against reduction and for a mechanistic model of interfield integration. According to the mechanistic model, different fields integrate their research by adding constraints on a multilevel description of a mechanism. Mechanistic integration may occur at a given level or in the effort to build a theory that oscillates among several levels. I develop this alternative model using (...) a putative exemplar of reduction in contemporary neuroscience: the relationship between the psychological phenomena of learning and memory and the electrophysiological phenomenon known as Long-Term Potentiation. A new look at this historical episode reveals the relative virtues of the mechanistic model over reduction as an account of interfield integration. (shrink)

Some claim that Non- reductive Physicalism is an unstable position, on grounds that NRP either collapses into reductive physicalism, or expands into emergentism of a robust or ‘strong’ variety. I argue that this claim is unfounded, by attention to the notion of a degree of freedom—roughly, an independent parameter needed to characterize an entity as being in a state functionally relevant to its law-governed properties and behavior. I start by distinguishing three relations that may hold between the degrees of freedom (...) needed to characterize certain special science entities, and those needed to characterize their composing physical entities; these correspond to what I call ‘reductions’, ‘restrictions’, and ‘eliminations’ in degrees of freedom. I then argue that eliminations in degrees of freedom, in particular—when strictly fewer degrees of freedom are required to characterize certain special science entities than are required to characterize their composing physical entities—provide a basis for making sense of how certain special science entities can be both physically acceptable and ontologically irreducible to physical entities. (shrink)

Lawrence Sklar in his book, Physics and Chance (1993), proposes a sophisticated account of reduction of thermodynamics (TD) by statistical mechanics (SM). I argue that Sklar's analysis of the alleged reduction of TD by SM is problematic in several respects. I consider a few counterexamples to show that none of what Sklar takes to be the central features of successful reduction in science (unification and identification) holds in the case of TD and SM. I suggest the broader conclusion that a (...) more useful way of understanding the relationship between TD and SM is as collaboration and competition among alternative methodologies rather than reduction of one theory to another. (shrink)

Reduction between theories in physics is often approached as an a priori relation in the sense that reduction is often taken to depend only on a comparison of the mathematical structures of two theories. I argue that such approaches fail to capture one crucial sense of “reduction,” whereby one theory encompasses the set of real behaviors that are well-modeled by the other. Reduction in this sense depends not only on the mathematical structures of the theories, but also on empirical facts (...) about where our theories succeed at describing real systems, and is therefore an a posteriori relation. (shrink)

Reduction and reductionism have been central philosophical topics in analytic philosophy of science for more than six decades. Together they encompass a diversity of issues from metaphysics and epistemology. This article provides an introduction to the topic that illuminates how contemporary epistemological discussions took their shape historically and limns the contours of concrete cases of reduction in specific natural sciences. The unity of science and the impulse to accomplish compositional reduction in accord with a layer-cake vision of the sciences, the (...) seminal contributions of Ernest Nagel on theory reduction and how they strongly conditioned subsequent philosophical discussions, and the detailed issues pertaining to different accounts of reduction that arise in both physical and biological science (e.g., limit-case and part-whole reduction in physics, the difference-making principle in genetics, and mechanisms in molecular biology) are explored. The conclusion argues that the epistemological heterogeneity and patchwork organization of the natural sciences encourages a pluralist stance about reduction. (shrink)

This chapter contains sections titled: Introduction: The Problem of Emergence and Reduction The Varieties of Reductionism: Ontological and Epistemological The Reduction and Emergence Debate Today: Specific Cases Seeming to Warrant the Label of Ontological or Epistemological Emergence Questions for Future Research.

A common belief is that evolution generally proceeds towards greater complexity at both the organismal and the genomic level, numerous examples of reductive evolution of parasites and symbionts notwithstanding. However, recent evolutionary reconstructions challenge this notion. Two notable examples are the reconstruction of the complex archaeal ancestor and the intron‐rich ancestor of eukaryotes. In both cases, evolution in most of the lineages was apparently dominated by extensive loss of genes and introns, respectively. These and many other cases of reductive evolution (...) are consistent with a general model composed of two distinct evolutionary phases: the short, explosive, innovation phase that leads to an abrupt increase in genome complexity, followed by a much longer reductive phase, which encompasses either a neutral ratchet of genetic material loss or adaptive genome streamlining. Quantitatively, the evolution of genomes appears to be dominated by reduction and simplification, punctuated by episodes of complexification. (shrink)

The aim of this article is to present a different perspective through which to examine reduction and emergence; namely, the perspective of chemistry’s relation to physics.

Most philosophical accounts of emergence are incompatible with reduction. Most scientists regard a system property as emergent relative to properties of the system's parts if it depends upon their mode of organization--a view consistent with reduction. Emergence can be analyzed as a failure of aggregativity--a state in which "the whole is nothing more than the sum of its parts." Aggregativity requires four conditions, giving tools for analyzing modes of organization. Differently met for different decompositions of the system, and in different (...) degrees, these conditions provide powerful evaluation criteria for choosing decompositions, and heuristics for detecting biases of vulgar reductionisms. This analysis of emergence is compatible with reduction. (shrink)

A belief common among philosophers and biologists alike is that Mendelian genetics has been or is in the process of being reduced to molecular genetics, in the sense of formal theory reduction current in the literature. The purpose of this paper is to show that there are numerous empirical and conceptual difficulties which stand in the way of establishing a systematic inferential relation between Mendelian and molecular genetics. These difficulties, however, have little to do with the traditional objections which have (...) been raised to reduction. (shrink)

Though most contemporary philosophers and scientists accept a physicalist view of mind, the recent surge of interest in the problem of consciousness has put the mind /body problem back into play. The physicalists' lack of success in dispelling the air of residual mystery that surrounds the question of how consciousness might be physically explained has led to a proliferation of options. Some offer alternative formulations of physicalism, but others forgo physicalism in favour of views that are more dualistic or that (...) bring in mentalistic features at the ground- floor level of reality as in pan-proto-psychism. My aim here is to give an overview of the recent philosophic discussion to serve as a map in locating issues and options. I will not offer a comprehensive survey of the debate or mark every important variant to be found in the recent literature. I will mark the principal features of the philosophic landscape that one might use as general orientation points in navigating the terrain. I will focus in particular on three central and interrelated ideas: those of emergence, reduction, and nonreductive physicalism. The third of these, which has emerged as more or less the majority view among current philosophers of mind, combines a pluralist view about the diversity of what needs to be explained by science with an underlying metaphysical commitment to the physical as the ultimate basis of all that is real. The view has been challenged from both left and right, on one side from dualists and on the other from hard core reductive materialists. Despite their differences, those critics agree in finding nonreductive physicalism an unacceptable and perhaps even incoherent position. They agree as well in treating reducibility as the essential criterion for physicality; they differ only about whether the criterion can be met. Reductive physicalists argue that it can, and dualists deny it. (shrink)

Of the various debates surrounding harm reduction, a conceptual tension that perhaps has the most relevance for the provision of services is that of harm reduction as a technical solution versus a contextualized social practice. The aim of this paper was to examine this conceptual tension. First, the two perspectives will be presented through the use of examples. Second, philosophical drivers that serve to underpin and justify each perspective will be explicated at the level of the knowledge that we privilege; (...) the ideologies that we subscribe to; and the interests that we stand to serve. In this paper, I argue that the existing tension between technical and social approaches to harm reduction is embedded within discord pertaining to ways of knowing, paradigms of inquiry, prevailing ideologies, and notions of harm and risk. Building on these sources of tension, I suggest a means of philosophical reconciliation between the two approaches and ways forward, namely through acknowledging multiple sources of knowledge, through embracing paradigmatic incommensurability, through considering alternative conceptions of people who use drugs as political subjects, through involving service providers and end‐users in shared decision‐making, and lastly through reaffirming people who use drugs as the intended beneficiaries of services. (shrink)

Is conceptual analysis required for reductive explanation? If there is no a priori entailment from microphysical truths to phenomenal truths, does reductive explanation of the phenomenal fail? We say yes . Ned Block and Robert Stalnaker say no.

One of the central problems in the philosophy of psychology is an updated version of the old mind-body problem: how levels of theories in the behavioral and brain sciences relate to one another. Many contemporary philosophers of mind believe that cognitive-psychological theories are not reducible to neurological theories. However, this antireductionism has not spawned a revival of dualism. Instead, most nonreductive physicalists prefer the idea of a one-way dependence of the mental on the physical.In Psychoneural Reduction, John Bickle presents a (...) new type of reductionism, one that is stronger than one-way dependency yet sidesteps the arguments that sank classical reductionism. Although he makes some concessions to classical antireductionism, he argues for a relationship between psychology and neurobiology that shares some of the key aims, features, and consequences of classical reductionism. Parts of Bickle's "new wave" reductionism have emerged piecemeal over the past two decades; this is his first comprehensive statement and defense of it to appear. (shrink)

The entropy-reduction hypothesis claims that the cognitive processing difficulty on a word in sentence context is determined by the word's effect on the uncertainty about the sentence. Here, this hypothesis is tested more thoroughly than has been done before, using a recurrent neural network for estimating entropy and self-paced reading for obtaining measures of cognitive processing load. Results show a positive relation between reading time on a word and the reduction in entropy due to processing that word, supporting the entropy-reduction (...) hypothesis. Although this effect is independent from the effect of word surprisal, we find no evidence that these two measures correspond to cognitively distinct processes. (shrink)

This is a survey article on reductive views of shared intention.

Unlike the overall framework of Ernest Nagel's work on reduction, his theory of intertheoretic connection still has life in it. It handles aptly cases where reduction requires complex representation of a target domain. Abandoning his formulation as too liberal was a mistake. Arguments that it is too liberal at best touch only Nagel's deductivist theory of explanation, not his condition of connectability. Taking this condition seriously gives a powerful view of reduction, but one which requires us to index explanatory power (...) to sciences as they are formulated at particular times. While we may thereby reduce more than philosophers have supposed, we must abandon hope (as Nagel did) of saying anything useful about reductionism. (shrink)

Grand debates over reduction and emergence are playing out across the sciences, but these debates have reached a stalemate, with both sides declaring victory on empirical grounds. In this book, Carl Gillett provides new theoretical frameworks with which to understand these debates, illuminating both the novel positions of scientific reductionists and emergentists and the recent empirical advances that drive these new views. Gillett also highlights the flaws in existing philosophical frameworks and reorients the discussion to reflect the new scientific advances (...) and issues, including the nature of 'parts' and 'wholes', the character of aggregation, and thus the continuity of nature itself. Most importantly, Gillett shows how disputes about concrete scientific cases are empirically resolvable and hence how we can break the scientific stalemate. Including a detailed glossary of key terms, this volume will be valuable for researchers and advanced students of the philosophy of science and metaphysics, and scientific researchers working in the area. (shrink)

This volume investigates the notion of reduction. Building on the idea that philosophers employ the term ‘reduction’ to reconcile diversity and directionality with unity, without relying on elimination, the book offers a powerful explication of an “ontological” notion of reduction the extension of which is (primarily) formed by properties, kinds, individuals, or processes. It argues that related notions of reduction, such as theory-reduction and functional reduction, should be defined in terms of this explication. Thereby, the book offers a coherent framework, (...) which sheds light on the history of the various reduction debates in the philosophy of science and in the philosophy of mind, and on related topics such as reduction and unification, the notion of a scientific level, and physicalism. (shrink)

Nagel’s official model of theory-reduction and the way it is represented in the literature are shown to be incompatible with the careful remarks on the notion of reduction Nagel gave while developing his model. Based on these remarks, an alternative model is outlined which does not face some of the problems the official model faces. Taking the context in which Nagel developed his model into account, it is shown that the way Nagel shaped his model and, thus, its well-known deficiencies, (...) are best conceived of as a mere by-product of his philosophical background. (shrink)

In this paper I examine Chalmers and Jackson’s defence of the a priori entailment thesis, that is, the claim that microphysical truths a priori entail ordinary non-phenomenal truths such as ‘water covers 60% of the Earth surface’, which they use as a premise for an argument against the possibility of a reductive explanation of consciousness. Their argument relies on a certain view about the possession conditions of macroscopic concepts such as WATER, known as ascriptivism. In the paper I distinguish two (...) versions of ascriptivism: reductive versus non-reductive ascriptivism. According to reductive ascriptivism, competent users of a concept have the ability to infer truths involving such concept from lower-level truths, whereas according to non-reductive ascriptivism, all that is required in order to be a competent user of a concept is to be able to infer truths involving that concept from other truths, which need not be lower-level truths. I argue, first, that the a priori entailment thesis is committed to reductive ascriptivism, and secondly, that reductive ascriptivism is problematic because it trivializes the notion of a priori knowledge. Therefore, I conclude that Chalmers and Jackson have not presented a convincing case for the claim that microphysical truths entail ordinary non-phenomenal truths a priori, especially when we understand this claim in the sense that is relevant for their argument against the possibility of a reductive explanation of consciousness. (shrink)

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic of (...) relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains. (shrink)

My aim here is threefold: to show that conceptual facts play a more significant role in justifying explanatory reductions than most of the contributors to the current debate realize; to furnish an account of that role, and to trace the consequences of this account for conceivability arguments about the mind.